101274: [AtCoder]ABC127 E - Cell Distance

Problem Statement

We have a grid of squares with $N$ rows and $M$ columns. Let $(i, j)$ denote the square at the $i$-th row from the top and $j$-th column from the left. We will choose $K$ of the squares and put a piece on each of them.

If we place the $K$ pieces on squares $(x_1, y_1)$, $(x_2, y_2)$, ..., and $(x_K, y_K)$, the cost of this arrangement is computed as:

$\sum_{i=1}^{K-1} \sum_{j=i+1}^K (|x_i - x_j| + |y_i - y_j|)$

Find the sum of the costs of all possible arrangements of the pieces. Since this value can be tremendous, print it modulo $10^9+7$.

We consider two arrangements of the pieces different if and only if there is a square that contains a piece in one of the arrangements but not in the other.

Constraints

• $2 \leq N \times M \leq 2 \times 10^5$
• $2 \leq K \leq N \times M$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$

Output

Print the sum of the costs of all possible arrangements of the pieces, modulo $10^9+7$.

Sample Input 1

2 2 2

Sample Output 1

8

There are six possible arrangements of the pieces, as follows:

• $((1,1),(1,2))$, with the cost $|1-1|+|1-2| = 1$
• $((1,1),(2,1))$, with the cost $|1-2|+|1-1| = 1$
• $((1,1),(2,2))$, with the cost $|1-2|+|1-2| = 2$
• $((1,2),(2,1))$, with the cost $|1-2|+|2-1| = 2$
• $((1,2),(2,2))$, with the cost $|1-2|+|2-2| = 1$
• $((2,1),(2,2))$, with the cost $|2-2|+|1-2| = 1$

The sum of these costs is $8$.

Sample Input 2

4 5 4

Sample Output 2

87210

Sample Input 3

100 100 5000

Sample Output 3

817260251

Be sure to print the sum modulo $10^9+7$.